It is often claimed in quantum gravity circles (I'm thinking of LQG in particular) that it is "not possible" to discretize spacetime. The issue, as I understand it, is that should one attempt to assign each point on a spatial lattice a value of e.g. the spatial metric $\gamma_{ab}$, the lattice spacings (both across time and space) will not be coordinate-invariants. This is pretty obviously true.

I work in numerical relativity, however, and we do precisely this all the time, maintaining parameters during the simulation such that the lattice spacings are indeed functions of spacetime. Should we seek to express quantities in a new "frame" we just transform those functions.

There appears to be no particular difficulty with this approach. One might, I suppose, run into trouble if a particular transformation were sought that brought previously irrelevant ultraviolet modes into the simulation domain, but this can be dealt with in a problem-dependent way by just choosing a high enough resolution at the outset. I am presumably, therefore, misunderstanding something about the difficulty the LQG people are getting at. What is it?